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Science and Hypothesis

Chapter 1: On the Nature of Mathematical Reasoning

Henri Poincaré

THE very possibility of mathematical science seems an insoluble contradiction. If this science is
only deductive in appearance, from whence is derived that perfect rigour which is challenged by
none? If, on the contrary, all the propositions which it enunciates may be derived in order by the
rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The
syllogism can teach us nothing essentially new, and if everything must spring from the principle
of identity, then everything should be capable of being reduced to that principle. Are we then to
admit that the enunciations of all the theorems

(2) with which so many volumes are filled, are only indirect ways of saying that A is A ?

No doubt we may refer back to axioms which are at the source of all these reasonings. If it is felt
that they cannot be reduced to the principle of contradiction, if we decline to see in them any
more than experimental facts which have no part or lot in mathematical necessity, there is still
one resource left to us: we may class them among à priori synthetic views. But this is no solution
of the difficulty — it is merely giving it a name; and even if the nature of the synthetic views had
no longer for us any mystery, the contradiction would not have disappeared; it would have only
been shirked. Syllogistic reasoning remains incapable of adding anything to the data that are
given it; the data are reduced to axioms, and that is all we should find in the
conclusions.

No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only
give us immediately evident truths borrowed from direct intuition; it would only be an
intermediary parasite. Should we not therefore have reason for asking if the syllogistic apparatus
serves only to disguise what we have borrowed?

The contradiction will strike us the more if we open any book on mathematics; on every page the
author announces his intention of generalising some proposition already known. Does the
mathematical method proceed front the particular to the general, and, if so, how can it be called
deductive ?

(3)

Finally, if the science of number were merely analytical, or could be analytically derived from a
few synthetic intuitions, it seems that a sufficiently powerful mind could with a single glance
perceive all its truths; nay, one might even hope that some day a language would be invented
simple enough for these truths to be made evident to any person of ordinary intelligence.

Even if these consequences are challenged, it must be granted that mathematical reasoning has of
itself a kind of creative virtue, and is therefore to be distinguished from the syllogism. The
difference must be profound. We shall not, for instance, find the key to the mystery in the
frequent use of the rule by which the same uniform operation applied to two equal numbers will
give identical results. All these modes of reasoning, whether or not reducible to the syllogism,
properly so called, retain the analytical character, and ipso facto, lose their power.

II.

The argument is an old one. Let us see how Leibnitz tried to show that two and two make four. I
assume the number one to be defined, and also the operation x+1, — i.e., the adding of unity to a
given number x. These definitions, whatever they may be, do not enter into the subsequent
reasoning. I next define the numbers 2, 3, 4 by the equalities: -

(1) 1+1=2; (2) 2+I=3; (3) 3+1= 4, and in

4) the same way I define the operation x+2 by the relation; (4) x+2 = (x+1)+1.

It cannot be denied that this reasoning is purely analytical. But if we ask a mathematician, he will
reply: "This is not a demonstration properly so called; it is a verification." We have confined
ourselves to bringing together one or other of two purely conventional definitions, and we have
verified their identity; nothing new has been learned. Verification differs from proof precisely
because it is analytical, and because it leads to nothing. It leads to nothing because the conclusion
is nothing but the premisses translated into another language. A real proof, on the other hand, is
fruitful, because the conclusion is in a sense more general than the premisses. The equality 2 + 2
= 4 can be verified because it is particular. Each individual enunciation in mathematics may be
always verified in the same way. But if mathematics could be reduced to a series of such
verifications it would not be a science. A chess-player, for instance, does not create a science by
winning a piece. There is no science but the science of the general. It may even be said that the
object of the exact sciences is to dispense with these direct verifications.

5)

III.

Let us now see the geometer at work, and try to surprise some of his methods. The task is not
without difficulty; it is not enough to open a book at random and to analyse any proof we may
come across. First of all, geometry must be excluded, or the question becomes complicated by
difficult problems relating to the rôle of the postulates, the nature and the origin of the idea of
space. For analogous reasons we cannot avail ourselves of the infinitesimal calculus. We must
seek mathematical thought where it has remained pure — i.e., in Arithmetic. But we still have to
choose ; in the higher parts of the theory of numbers the primitive mathematical ideas have
already undergone so profound an elaboration that it becomes difficult to analyse them.

It is therefore at the beginning of Arithmetic that we must expect to find the explanation we seek;
but it happens that it is precisely in the proofs of the most elementary theorems that the authors
of classic treatises have displayed the least precision and rigour. We may not impute this to them
as a crime; they have obeyed a necessity. Beginners are not prepared for real mathematical
rigour; they would see in it nothing but empty, tedious subtleties. It would be waste of time to try
to make them more exacting; they have to pass rapidly and without stopping over the road

(6) which was trodden slowly by the founders of the science.

Why is so long a preparation necessary to habituate oneself to this perfect rigour, which it would
seem should naturally be imposed on all minds ? This is a logical and psychological problem
which is well worthy of study. But we shall not dwell on it ; it is foreign to our subject. All I wish
to insist on is, that we shall fail in our purpose unless we reconstruct the proofs of the elementary
theorems, and give them, not the rough form in which they are left so as not to weary the
beginner, but the Form which will satisfy the skilled geometer.

DEFINITION OF ADDITION.

I assume that the operation x+ 1 has been defined; it consists in adding the number 1 to a given
number x. Whatever may be said of this definition, it does not enter into the subsequent
reasoning.

We now have to define the operation x + a, which consists in adding the number a to any given
number x. Suppose that we have defined the operation x + (a - 1) ; the operation x + a will be
defined by the equality: (1) x+a= [x+ (a - 1)]+ 1. We shall know what x +a is when we know
what x + (a - 1) is, and as I have assumed that to start with we know what x + 1 is, we can define
successively and " by recurrence" the operations x + 2, x + 3, etc. This definition deserves a
moment's

(7) attention; it is of a particular nature which distinguishes it even at this stage from the purely
logical definition; the equality (1), in fact, contains an infinite number of distinct definitions,
each having only one meaning when we know the meaning of its predecessor.

PROPERTIES OF ADDITION.

Associative. — I say that a+(b+c) = (a+b)+c; in fact, the theorem is true for c=1. It may then be
written a+(b+1)=(a+b)+1; which, remembering the difference of notation, is nothing but the
equality (1) by which I have just defined addition. Assume the theorem true for c= y, I say that it
will be true for c=y+ 1. Let (a+b)+ =a+(b+), it follows that [(a+b)+] +1= [a+(b+ )] +1;
or by def. (1) — (a+b) + (+1) =a + (b + + 1)=a + [b+ (+ 1)], which shows by a series of
purely analytical deductions that the theorem is true for + 1. Being true for c = 1, we see that it
is successively true for s` c = 2, c = 3, etc.

Commutative.— (1) I say that a + 1 = 1 + a. The theorem is evidently true for a = 1 ; we can
verify by purely analytical reasoning that if it is true for a= it will be true for a=y+ 1. [1]Now,it is true for a =1, and therefore is true for a=2, a=3, and so on. This is what is meant by saying
that the proof is demonstrated " by recurrence."

I say that a + b = b + a. The theorem has just

8) been shown to hold good for b = 1, and it may be verified analytically that if it is true for
b=ß, it will be true for b=ß+ 1. The proposition is thus established by recurrence.

DEFINITION OF MULTIPLICATION.

We shall define multiplication by the equalities: (I) a x 1 =a. (2) a x b= [a x (b- 1)] + a. Both of
these include an infinite number of definitions; having defined a x I, it enables us to define in
succession a x 2, a x3, and so on.

PROPERTIES OF MULTIPLICATION.

Distributive. — I say that (a+b) x c = (a x c) + (b x c). We can verify analytically that the
theorem is true for c=1; then if it is true for c=, it will be true for c= + 1. The proposition is
then proved by recurrence.

Commutative. — (1) Isay that a x 1= 1 x a. The theorem is obvious for a=1. We can verify
analytically that if it is true for a=, it will be true for a=+I.

(2) I say that a x b=b xa. The theorem has just been proved for b=1. We can verify analytically
that if it be true for b= it will be true for b= + I .

IV.

This monotonous series of reasonings may now be laid aside; but their very monotony brings
vividly to light the process, which is uniform,

(9) and is met again at every step. The process is proof by recurrence. We first show that a
theorem is true for n= 1 ; we then show that if it is true for n- 1 it is true for it, and we conclude
that it is true for all integers. We have now seen how it may be used for the proof of the rules of
addition and multiplication — that is to say, for the rules of the algebraical calculus. This
calculus is an instrument of transformation which lends, itself to many more different
combinations than the simple syllogism; but it is still a purely analytical instrument, and is
incapable of teaching us anything new. If mathematics had no other instrument, it would
immediately be arrested in its development; but it has recourse anew to, the same process
— i.e.,
to reasoning by recurrence, and it can continue its forward march. Then if we look carefully, we
find this mode of reasoning at every step, either under the simple form which we have just given
to it, or under a more or less modified form. It is therefore mathematical reasoning par
excellence, and we must examine it closer.

V.

The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak,
in a single formula, an infinite number of syllogisms. We shall see this more clearly if we
enunciate the syllogisms one after another. They follow one another, if one may use the
expression, in a cascade. The following are the hypothetical

(10) syllogisms: — The theorem is true of the number 1. Now, if it is true of 1, it is true of 2;
therefore it is true of 2. Now, if it is true of 2, it is true of 3; hence it is true of 3, and so on. We
see that the conclusion of each syllogism serves as the minor of its successor. Further, the majors
of all our syllogisms may be reduced to a single form. If the theorem is true of n - 1, it is true of
n.

We see, then, that in reasoning by recurrence we confine ourselves to the enunciation of the
minor of the first syllogism, and the general formula which contains as particular cases all the
majors. This unending series of syllogisms is thus reduced to a phrase of a few lines.

It is now easy to understand why every particular consequence of a theorem may, as I have above
explained, be verified by purely analytical processes. If, instead of proving that our theorem is
true for all numbers, we only wish to show that it is true for the number 6 for instance, it will be
enough to establish the first five syllogisms in our cascade. We shall require 9 if we wish to
prove it for the number 10; for a greater number we shall require more still; but however great
the number may be we shall always reach it, and the analytical verification will always be
possible. But however far we went we should never reach the general theorem applicable to all
numbers, which alone is the object of science. To reach it we should require an infinite number
of syllogisms, and we should have to cross an abyss

(11) which the patience of the analyst, restricted to the resources of formal logic, will never
succeed in crossing.

I asked at the outset why we cannot conceive of a mind powerful enough to see at a glance the
whole body of mathematical truth. The answer is now easy. A chess-player can combine for four
or five moves ahead; but, however extraordinary a r player he may be, he cannot prepare for more
than a finite number of moves. If he applies his faculties to Arithmetic, he cannot conceive its
general truths by direct intuition alone; to prove even the smallest theorem he must use reasoning
by recurrence, for that is the only instrument which e enables us to pass from the finite to the
infinite. This instrument is always useful, for it enables us to leap over as many stages as we
wish; it frees us from the necessity of long, tedious, and monotonous verifications which would
rapidly become impracticable. Then when we take in hand the general theorem it becomes
indispensable, for otherwise we should ever be approaching the analytical verification without
ever actually reaching it. In this domain of Arithmetic we may think ourselves very far from the
infinitesimal analysis, but the idea of mathematical infinity is already playing a preponderating
part, and without it there would be no science at all, because there would be nothing general.

(12)

VI.

The views upon which reasoning by recurrence is based may be exhibited in other forms; we may
say, for instance, that in any finite collection of different integers there is always one which is
smaller than any other. We may readily pass from one enunciation to another, and thus give
ourselves the illusion of having proved that reasoning by recurrence is legitimate. But we shall
always be brought to a full stop —we shall always come to an indemonstrable axiom, which will
at bottom be but the proposition we had to prove translated into another language. We cannot
therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the
principle of contradiction. Nor can the rule come to us from experiment. Experiment may teach
us that the rule is true for the first ten or the first hundred numbers, for instance; it will not bring
us to the indefinite series of numbers, but only to a more or less long, but always limited, portion
of the series.

Now, if that were all that is in question, the principle of contradiction would be sufficient, it
would always enable us to develop as many syllogisms as we wished. It is only when it is a
question of a single formula to embrace an infinite number of syllogisms that this principle
breaks down, and there, too, experiment is powerless to aid. This rule, inaccessible to analytical
proof and to experiment, is the exact type of the à priori

(13) synthetic intuition. On the other hand, we cannot see in it a convention as in the case of the
postulates of geometry.

Why then is this view imposed upon us with such an irresistible weight of evidence ? It is
because it is only the affirmation of the power of the mind which knows it can conceive of the
indefinite repetition of the same act, when the act is once possible. The mind has a direct
intuition of this power, and experiment can only be for it an opportunity of using it, and thereby
of becoming conscious of it.

But it will be said, if the legitimacy of reasoning by recurrence cannot be established by
experiment alone, is it so with experiment aided by induction? We see successively that a
theorem is true of the number 1, of the number 2, of the number 3, and so on —
the law is
manifest, we say, and it is so on the same ground that every physical law is true which is based
on a very large but limited number of observations.

It cannot escape our notice that here is a striking analogy with the usual processes of induction.
But an essential difference exists. Induction applied to the physical sciences is always uncertain,
because it is based on the belief in a general order of the universe, an order which is external to
us. Mathematical induction — i.e., proof by recurrence — is, on the contrary, necessarily
imposed on us, because it is only the affirmation of a property of the mind itself.

(14)

VII.

Mathematicians, as I have said before, always endeavour to generalise the propositions they have
obtained. To seek no further example, we have just shown the equality, a + 1 = 1 + a, and we
then used it to establish the equality, a + b = b + a, which is obviously more general.
Mathematics may, therefore, like the other sciences, proceed from the particular to the general.
This is a fact which might otherwise have appeared incomprehensible to us at the beginning of
this study, but which has no longer anything mysterious about it, since we have ascertained the
analogies between proof by recurrence and ordinary induction.

No doubt mathematical recurrent reasoning and physical inductive reasoning are based on
different foundations, but they move in parallel lines and in the same direction—namely, from the
particular to the general.

Let us examine the case a little more closely. To prove the equality a + 2 = 2 + a ... ... (1), we
need only apply the rule. a + 1 = 1 + a, twice, and write

a+2=a+1+1=1+a+1=1+1+a=2+a . (2).

The equality thus deduced by purely analytical means is not, however, a simple particular case. It
is something quite different. We may not therefore even say in the really analytical and deductive
part of mathematical reasoning that we proceed from the general to the particular in the ordinary
sense of the words. The two sides of

(15) the equality (2) are merely more complicated combinations than the two sides of the
equality (1), and analysis only serves to separate the elements which enter into these
combinations and to study their relations.

Mathematicians therefore proceed "by construction," they "construct" more complicated
combinations. When they analyse these combinations, these aggregates, so to speak, into their
primitive elements, they see the relations of the elements and deduce the relations of the
aggregates themselves. The process is purely analytical, but it is not a passing from the general to
the particular, for the aggregates obviously cannot be regarded as more particular than their
elements.

Great importance has been rightly attached to this process of " construction," and some claim to
see in it the necessary and sufficient condition of the progress of the exact sciences. Necessary,
no doubt, but not sufficient ! For a construction to be useful and not mere waste of mental effort,
for it to serve as a stepping-stone to higher things, it must first of all possess a kind of unity
enabling us to see something more than the juxtaposition of its elements. Or more accurately,
there must be some advantage in considering the construction rather than the elements
themselves. What can this advantage be ? Why reason on a polygon, for instance, which is
always decomposable into triangles, and not on elementary triangles' It is because there are
properties of

(16) polygons of any number of sides, and they can be immediately applied to any particular kind
of polygon. In most cases it is only after long efforts that those properties can be discovered, by
directly studying the relations of elementary triangles. If the quadrilateral is anything more than
the juxtaposition of two triangles, it is because it is of the polygon type.

A construction only becomes interesting when it can be placed side by side with other
analogous constructions for forming species of the same genus. To do this we must necessarily
go back from the particular to the general, ascending one or more steps. The analytical process "
by construction" does not compel us to descend, but it leaves us at the same level. We can only
ascend by mathematical induction, or from it alone can we learn something new. Without the aid
of this induction, which in certain respects differs from, but is as fruitful as, physical induction,
construction would be powerless to create science.

Let me observe; in conclusion, that this induction is only possible if the same operation can be
repeated indefinitely. That is why the theory of chess can never become a science, for the
different moves of the same piece are limited and do not resemble each other.

Notes

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